Rational numbers

Rational numbers are all the numbers that can be written as the quotient of two integers, that is, in the form of a fraction.

Definition

Rational numbers are all the numbers that can be written in the form

$\frac{a}{b} a, b \in Z, b \neq 0.$

The condition that both numbers are integers has an obvious purpose. If you picked a number that is not rational, for example π, and built a fraction like this:

$\frac{π}{1},$

you would get π back, which is not rational. That is why the numerator and the denominator must both be integers. And since we cannot divide by zero, the denominator must be non-zero.

We can also say that rational numbers are the numbers with a terminating decimal expansion and the numbers with an infinite decimal expansion in which some part repeats periodically. So here are examples of rational numbers, first in the form of fractions:

$\frac{1}{3}, - \frac{5}{9}, \frac{13}{29}, \frac{4}{1}, \dots$

and then numbers in the form of decimals: 0.1; −5; 14.5; −12.93; 0.33333…= $0, \overset{―}{3}$

Recurring decimals

In the previous example we had the number 0.3333… This is the decimal form of the fraction 1/3, and it has an infinite decimal expansion. Even so, it is rational, because the expansion is recurring. A recurring decimal means that from some point on, the same sequence of digits repeats for ever. In our case the 3 repeated for ever. But the sequence of digits 12345 could just as well repeat for ever, and it would still be a recurring decimal, a rational number.

So every number with a terminating decimal expansion is rational. Every number with an infinite decimal expansion in which some part recurs is also rational. A number with an infinite decimal expansion in which no part recurs is an irrational number.

The recurring digits are marked either with three dots or, more often, with a bar over the digits that repeat. A single digit can recur, but so can several digits. The recurring part can start right at the beginning of the decimal part or anywhere later. These are all recurring decimals:

$\begin{array}{rclcl} 7 / 3 & = & 2.33333 \dots & = & 2, \overset{―}{3} \\ 16 / 11 & = & 1.454545 \dots & = & 1, \overset{―}{45} \\ 11 / 6 & = & 1.833333 \dots & = & 1.8 \overset{―}{3} \end{array}$

Notation and meaning

We write the rational numbers with the letter Q with a double curve: ℚ. It comes from the word “quotient”, which means the result of a division.

Rational numbers are used to describe a part of a whole, typically “a half” or “a tenth”. These are parts of a whole that can be expressed as the quotients 1/2 and 1/10. In such cases the denominator stands for the whole and the numerator for the part. If the numerator equals the denominator, we have the whole thing. These amounts are usually converted to percentages.

Properties

The rational numbers form an infinite countable set.

The rational numbers are closed under addition, subtraction, multiplication and division. This means that when we divide two rational numbers, we get a rational number again. This is a change from the integers, which were not closed under division.
The rational numbers contain all the integers. If every rational number can be written as a fraction a/b, then it is enough to take b = 1, and after dividing, a/1 = a always gives us back the value of the numerator.
If we take two rational numbers a and b with a < b, we can always find another rational number q with a < q < b. In other words, between any two rational numbers we can always find another rational number. A stronger version holds too: between any two rational numbers there are infinitely many other rational numbers.
A consequence of the previous statement is that there is no smallest positive (or largest negative) rational number. We can use proof by contradiction. Let m be the smallest positive rational number. Then we should not be able to find a smaller positive rational number. However, by the previous statement 0 < m, and so there is a rational number s with 0 < s < m. We have found a number s that is greater than zero (it is positive) and smaller than m. This contradicts the assumption that m is the smallest positive rational number.

Operations with rational numbers

All the usual basic operations are described in the article on fractions.

Converting a recurring decimal to a fraction

Since a recurring decimal is also a rational number, it must be possible to convert it to a fraction. Let’s see how. To start, take the recurring decimal a = 0.333… We write

$a = 0, \overset{―}{3}$

Now we carry out the usual equivalent manipulations of equations and multiply the whole equation by ten:

$10 a = 3, \overset{―}{3}$

On the right-hand side a 3 has moved into the units place, and after the decimal point there are still infinitely many 3s. Now we subtract a from the equation, that is, we subtract 0.333…, which gets rid of the infinite expansion.

$9 a = 3$

That step may have been a little tricky, so let me write it out. We carried out this operation:

$\begin{array}{ccccccc} 3 & , & 3 & 3 & 3 & \dots \\ - & 0 & , & 3 & 3 & 3 & \dots \\ = & 3 & , & 0 & 0 & 0 & \dots \end{array}$

We just divide the equation 9a = 3 by nine and get the result:

$a = \frac{3}{9} = \frac{1}{3}$

Let’s try one more example:

$a = 1.454545 \dots = 1, \overset{―}{45}$

The method will differ slightly. In the previous example the recurring part had length one. More precisely, it could have been as long as we liked, because the same digits kept repeating, but it suited us to take it as length one. Now the recurring part has length two. Again, we could take a length of six, but it does not suit us. To subtract the whole recurring part, this time we have to multiply by one hundred:

$100 a = 145, \overset{―}{45}$

We subtract one a:

$99 a = 144$

And divide by 99:

$a = \frac{144}{99} = \frac{16}{11}$

If you try dividing these numbers on a calculator, you will get exactly 1.454545… Most calculators round, however, so yours may show a slightly different result.

An interesting case: 0.9999…

Just for fun, let’s convert the recurring decimal

$a = 0, \overset{―}{9}$

to a fraction. Multiply by ten:

$10 a = 9, \overset{―}{9}$

Subtract a:

$9 a = 9$

Divide by nine:

$a = 1$

So we have got 1, which means that the number 0.999… is equal to 1.